It can be proven[1]} that the so-obtained LMOs change least (in the least-squares sense) from the pFLMOs and can therefore be termed FLMOs[2]}.
The very key for the success of F2M is that the pFLMOs are local not only in space by construction but also in energy in the sense that
their occupation numbers do not deviate significantly from the ideal value 2, 1 or 0.
Different partitions of the same molecule may affect the locality of the FLMOs but only to a minor extent,
which is tolerable given the dramatic gain in efficiency. Note in passing that, at variance with this noniterative “top-down least-change”
(“diagonalize-then-block-diagonalize”) algorithm, an iterative “bottom-up least-change” algorithm is also available, which does not invoke the global CMOs at all[3]}.
